• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,347 answers , 22,726 comments
1,470 users with positive rep
818 active unimported users
More ...

  Does the shortest path between two braids pass through string links?

+ 10 like - 0 dislike

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.

This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, are related by ambient isotopy if and only if they are related by height-preserving ambient isotopy. This is a non-trivial fact whose proofs are all somewhat complicated (Stallings Theorem/ Magnus expansion/ embedding fibrations).

Given that there is no theoretical advantage to injecting braids into string links (distinct braids stay distinct), I wonder whether there is a computational advantage in doing so. Explicitly, given diagrams for $B_1$ and for $B_2$, is the minimum number of Reidemeister moves between them always realized for Reidemeister moves between braids? Or might the `shortest path between two braids' pass through string links?

Question: Is there an example of a pair of equivalent braid diagrams, considered as tangle diagrams, such that the minimum number of Reidemeister moves between them is increased if we allow only braid-like Reidemeister moves (i.e. if the result of each Reidemeister move must also be a braid)?

This post imported from StackExchange MathOverflow at 2014-10-25 10:37 (UTC), posted by SE-user Daniel Moskovich

asked Feb 16, 2014 in Mathematics by Daniel Moskovich (130 points) [ revision history ]
retagged Oct 31, 2014 by dimension10
Most voted comments show all comments

"It doesn't seem to be related to physics" ... wrong! Braids, knots, etc. are some of the fundamental tools using which many concepts in modern condensed matter and high energy physics can be adequately expressed. Yes, it would be nice to have a physics-related motivation for the question, but given the growing significance of braids (and not because of my own personal bias) for modern physics, such questions should, IMO, be considered useful for a physics audience.

I think that you would be right if everybody was importing such questions without any obvious physics reference. Despite that, we can be flexible sometimes and maybe ask the user to state the relevance to physics (maybe)? In any case, the particular question is not unrelevant.

Maybe in the mid-term exam ;). Actually, this is what I meant, that it should be the user's who is making the question job to give the physical background/motivation. In the meanwhile, and since I am not* an expert on the subject let me suggest Baez's work on higher dimensional Lie algebras and the relation of braidings with the scattering amplitudes in 2+1 dimensions. I have made some relevant question (not on the braids but related to them) before but got no answer. Especially after such a discussion maybe Daniel should do so.

@conformal_gk, but it's not imported by the author, Daniel probably is not aware that this post is here. Although I know Daniel in person, it would feel a bit weird if I ask him to "defend" a post that he never intended to put here.

Yiyang invited me to join Physics Overflow... so now I should comment here.

Much quantum topology for knots and links (e.g. the Jones polynomial) factors through Markov's Theorem, which relates the theory of braid groups (essentially algebra) with the theory of knots and links (low dimensional topology) by providing necessary and sufficient conditions for two different braids to have closures which represent ambient isotopic topological objects in 3 dimensions. Braid groups live in a world with much less structure than low dimensional topology.

Essentially, I'm specifically asking whether the injection of braids into string links leads to speedups (thus whether it is also an "injection of complexities"), in the way that injecting integers into real numbers leads to speedup proofs. I'm trying to understand an aspect of in what sense the sequence "first this crossing, then that crossing" that we have in a braid, and that occurs in the statistical mechanics context (or in the Witten TQFT context) from which the Jones polynomial arises is fundamental. Hiding behind this is the vague question of how fundamental the temporal order of things in TQFT (the "time axis", "Morse function", or whatever) really is, and whether there is something concrete to be gained by dropping it.

My vague intuition is that the answer to the above question is probably "no", but that there is a close variant for which the answer is "yes".
Most recent comments show all comments

@conformal_gk, I don't disagree that knot theory can be closely related to physics, but it doesn't mean every question in knot theory is(I can be wrong on this particular question, but my point still stands). It is somewhat psychologically unsettling that just because a broad category is related to physics, any (grad level) question in that category can be imported.

@conformal_gk, I'd surely love to hear how it is related.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights